Even if the Luftwaffe had gained air superiority the Kriegsmarine could never have sustained an invasion across the channel in the face of the overwhelming numerical superiority of the Royal Navy.
By the end of August the RN had 3 battleships, a carrier, 8 cruisers and 76 destroyers specifically stationed ready to destroy any invasion fleet. The most distant heavy units were based on the Firth of Forth and could have reached the Dover Straits within 20hrs.
Despite this commitment to home defence greater naval resources still were retained in the north at Scapa Flow and further afield in the Mediterranean.
There can be no doubt, that even in the face of heavy attacks by the Luftwaffe, the RN would have intervened decisively.
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